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\def\startproof{\noindent{\sl Proof.}\enspace}
\def\endproof{{\unskip\nobreak\hfil\penalty50\hskip1em
  \hbox{}\nobreak\hfil$\blacksquare$%
    \parfillskip=0pt \finalhyphendemerits=0 \par}}


%\def\endproof{ $\blacksquare$}

% put equation numbers on the left instead of on the right
\def\leqno(#1){\eqno{\hbox to .01pt{}\rlap{\rm \hskip -\displaywidth(#1)}}}


%  This makes the paper look better
          \parskip\smallskipamount
%   in my humble opinion


% Numbering references

\newcount\refCount
\def\newref#1 {\advance\refCount by 1
\expandafter\edef\csname#1\endcsname{\the\refCount}}

  \newref ANTIV %Antwerp IV, Birch and Kuyk, eds.
  \newref GRBU %Gross and Buhler
  \newref CHI % Chi's thesis
  \newref DELCORVALIS % Deligne Conjectures
  \newref FALT %Faltings in German
  \newref FW  % Faltings-Wust
  \newref BHG %Gross's thesis
  \newref SGA % Grothendieck
  \newref HIDA % Hida Am J article
  \newref KOIKE % on 3^4 and 3^5
  \newref LORENZ % Dino's paper
  \newref GAD %Gadfly paper
  \newref MILNE % Milne IM
  \newref MUMF % Abelian varieties book by Mumford
  \newref OGG %elliptic curves and wild ram
  \newref ANNALS % my annals paper
  \newref MYTHESIS % my American Journal article
  \newref BONN %my Bonn article
  \newref MATHANN % Mathematische Annalen paper
  \newref DPP % my DPP paper from 1979--80
  \newref MG % abelian l-adic reps book
  \newref SERREINV % Serre 1972 inventiones article
  \newref JPSDURHAM %Serre at Durham
  \newref DUKE   % Serre's Duke article
  \newref DISCONT %Shimura 1967
  \newref PINK %Sh's pink paper
  \newref SHIM %Shimura's book
  \newref CFHO %Shimura on class fields and Hecke operators
  \newref FACTOR % On the factors..
  \newref BOUNDED %Sh Math An, bounded symmetric domains
  \newref SHTAN % Shimura-Taniyama
  \newref WEILFD
  \newref WEIL %adeles and algebraic groups


% math macros


 \let\tempcirc=\circ
 \def\circ{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}}
      % The above two lines make "circ" smaller (I prefer it this way)

 \def\seteq{\mathrel{:=}}
 \def\GalQ{\Gal(\Qbar/\Q)}
 \def\GalQq{\Gal(\Qqbar/\Qq)}
 \def\Aut{\mathop{\roman{Aut}}\nolimits}
 \def\deg{\mathop{\roman{deg}}\nolimits}
 \def\Br{\mathop{\roman{Br}}\nolimits}
 \def\Sym{\mathop{\roman{Sym}}\nolimits}
 \def\Supp{\mathop{\roman{Supp}}\nolimits}
 \def\Gal{\mathop{\roman{Gal}}\nolimits}
 \def\Res{\mathop{\roman{Res}}\nolimits}
 \def\End{\mathop{\roman{End}}\nolimits}
 \def\Lie{\mathop{\roman{Lie}}\nolimits}
 \def\Hom{\mathop{\roman{Hom}}\nolimits}
 \def\Tr{\mathop{\roman{Tr}}\nolimits}
 \def\SL{\mathop{\bold{SL}}\nolimits}
 \def\PSL{\mathop{\bold{PSL}}\nolimits}
 \def\GL{\mathop{\bold{GL}}\nolimits}
 \def\ord{\mathop{\roman{ord}}\nolimits}
 \def\Norm{\mathop{\bold N}\nolimits}
 \def\GLTWO{$\GL_2$}
 \def\SLTWO#1{\SL(2,#1)}
 \def\PSLTWO#1{\PSL(2,#1)}
 \def\trace{\mathop{\roman{tr}}\nolimits}
 \def\Pico{\mathop{\roman{Pic}}\nolimits ^{\roman o}}
 \def\ZNZ{(\Z/N\Z)^*}
 \def\Q{{\bold Q}}
 \def\R{\bold R}
 \def\N{{\Cal N}}
 \def\Qp{\Q_p}
 \def\Qell{\Q_\ell}
 \def\Qellbar{\overline{\Q}_\ell}
 \def\Zell{\Z_\ell}
 \def\C{{\bold C}}
 \def\Qbar{\overline\Q}
 \def\Qq{\Q_q}
 \def\Zq{\Z_q}
 \def\Qqbar{{\overline\Q}_q}
 \def\F{{\bold F}}
 \def\Fbar{\overline\F}
 \def\Fp{{\bold F}_p}
 \def\Fq{{\bold F}_q}
 \def\Fell{{\F_\ell}}
 \def\Zell{\Z_\ell}
 \def\Fpbar{\overline{\F}_p}
 \def\Fqbar{\overline{\F}_q}
 \def\rholambdabar{\overline{\rho}_\lambda}
 \def\ssigma{{}^\sigma\mkern-\thinmuskip}
 \def\ttau{{}^\tau\mkern-\thinmuskip}
 \def\ggg{{}^g\mkern-\thinmuskip}
 \def\Flambdabar{\overline{\F}_\lambda}
 \def\chiellbar{\overline{\chi}_\ell}
 \def\sigmabar{{\bar{\sigma}}}
 \def\Z{{\bold Z}}
 \def\T{{\bold T}}
 \def\Frob{{\roman{Frob}}}
 \def\Xo#1{X_o(#1)}  % ditto
 \let\phi\varphi

\topmatter

 \title
Abelian varieties over $\Q$ and modular forms
 \endtitle

 \author
Kenneth A. Ribet
 \endauthor

 \affil
University of California, Berkeley
 \endaffil
\rightheadtext{Abelian varieties over $\Q$ and modular forms}
\address{UC Mathematics Department, Berkeley, CA  94720 USA}
\endaddress

\email ribet\@math.berkeley.edu \endemail


  \thanks 
This manuscript was written in conjunction with the author's
lectures at the seventh KAIST mathematics workshop in 
Taejon, Korea (August 11--14, 1992).  The author describes
 work which
was begun during his visits to the Hebrew University
of Jerusalem and the Universit\'e de Paris~XI (Orsay).  It
is a pleasure to thank these institutions for their
hospitality.
The author wishes to thank J-P.~Serre for suggesting the
problem and for pointing out the relevance of Tate's
theorem (Theorem~6.3 below.) 
The author was supported in part by NSF Grant \#DMS 88-06815.
  \endthanks


\endtopmatter


\document

\head 1. Introduction.\endhead \noindent
Let $C$ be an elliptic curve over~$\Q$.  
Let $N$ be the conductor of~$C$.
The Taniyama conjecture
asserts that there is a non-constant map of algebraic curves
$\Xo N \to C$ which is defined over~$\Q$. 
Here, $\Xo N$
is the standard modular curve associated with the problem
of classifying elliptic curves $E$ together
with cyclic subgroups of~$E$ having
order~$N$.  

The Taniyama  conjecture may be reformulated in various
ways.  For example, let $X_1(M)$ be the modular curve
associated with the problem of classifying elliptic curves
$E$ along with a point of order~$M$ on~$E$.
One knows that
if there is a non-constant map
$X_1(M) \to C$ over~$\Q$ for some~$M\ge1$, then there is a
non-constant map $\Xo N\to C$.  (For a result in this
direction, see \cite\LORENZ.)

In a recent article \cite\GAD, B.~Mazur introduced another
type of reformulation.
Mazur says that $C$ possesses a ``hyperbolic uniformization
of arithmetic type" if there is a non-constant map
over the complex field $\pi\:X_1(M)_{\C}\to C_{\C}$ for some
$M\ge1$.  The map~$\pi$
is not required to be defined over~$\Q$.
In \cite{\GAD, Appendix},
Mazur proves: Suppose that
there is a non-constant map $\pi\:X_1(M)_{\C}\to C_{\C}$.
Then there is a non-constant map $\pi'\:X_1(M') \to C$
over~$\Q$, where $M'$ is a suitable positive integer
(which may be different from~$M$).
As mentioned above,
it follows easily from
the existence of~$\pi'$ that $C$ satisfies Taniyama's conjecture.  
We thus arrive at the following conclusion:

\proclaim{(1.1) Theorem {\rm [Mazur]}}
Let $C$ be an elliptic curve over~$\Q$.  Then $C$
satisfies Taniyama's conjecture if and only if
$C$ admits
a hyperbolic uniformization of arithmetic type over~$\C$.
\endproclaim

In connection with \cite\GAD, Serre asked for a conjectural
characterization of elliptic curves over $\C$ which 
admit a hyperbolic uniformization of arithmetic type.  All
such curves are defined over~$\Qbar$, the algebraic closure
of~$\Q$ in~$\C$.  Thus one wishes to characterize
those elliptic curves over~$\Qbar$ which are quotients of some
modular curve $X_1(M)$.
Equivalently, one wishes to study the set of elliptic
curves
which are quotients of the Jacobian $J_1(N)_{\Qbar}$
of~$X_1(N)_{\Qbar}$, for some $N\ge1$.
It is well known that all complex multiplication elliptic
curves have this property; this follows from \cite{\PINK,
Th.~1}.
Hence we are principally interested in elliptic curves
$C/\Qbar$ without complex multiplication.

In this article, we introduce the concept of an 
abelian variety over~$\Q$ which is of ``\GLTWO-type."
Roughly speaking, this is an abelian variety over~$\Q$
whose algebra of $\Q$-endomorphisms is a
number field of degree equal to the dimension of the
abelian variety.  It is easy to see that $J_1(N)$ decomposes
up to isogeny over~$\Q$ as a product of such abelian
varieties.  Hence any $C/\Qbar$ which is a quotient of some
$J_1(N)$ is a quotient (over~$\Qbar$)
of an abelian variety of \GLTWO-type over~$\Q$.

We prove in this
article the following facts (the third is a
simple corollary of the first two):
%\medskip\noindent
\item{1.} Let $C$ be an elliptic curve over~$\Qbar$
without complex multiplication.  Then
$C$ is a quotient of an abelian variety of~\GLTWO-type over
$\Q$ if and only if $C$ is isogenous to each of its Galois
conjugates $\ssigma C$ with $\sigma\in\GalQ$.
(If $C$ has this latter property, we say that $C$
is a ``$\Q$-curve."  The terminology is borrowed from
Gross~\cite\BHG.)
%\medskip\noindent
\item{2.} Assume Serre's
conjecture \cite{\DUKE, (3.2.4$_?$)}
on representations of~$\GalQ$.
Then every abelian variety of \GLTWO-type over~$\Q$ is a
quotient of~$J_1(N)$ for some $N\ge1$.
%\medskip\noindent
\item{3.} Let $C/\Qbar$ be an elliptic curve without complex
multiplication.  If $C$ is a quotient of $J_1(N)_{\Qbar}$ for
some $N$, then $C$ is a $\Q$-curve.
Conversely, suppose that $C$ is a $\Q$-curve.  Then
if the conjecture \cite{\DUKE, (3.2.4$_?$)} is correct, $C$
is a quotient of~$J_1(N)_{\Qbar}$ for some~$N$.

\noindent
In summary, we arrive at a conjectural characterization of
elliptic curves over~$\Qbar$ which are quotients of
modular curves $X_1(N)$: they are exactly the 
$\Q$-curves in the sense indicated above.
This characterization was predicted by~Serre.
\head 2.
Abelian varieties over $\Q$ of \GLTWO-type.\endhead \noindent
%
We will be concerned with abelian varieties over~$\Q$ which
admit actions of number fields that are ``as large as possible."
We first quantify this concept.

Suppose that $A$ is an abelian variety over~$\Q$ and that $E$
is a number field acting on $A$ up to isogeny
over~$\Q$:
$$ E \hookrightarrow \Q \otimes_\Z \End_\Q(A).$$
By functoriality, $E$ acts on the space of tangent vectors
$\Lie(A/\Q)$, which is a $\Q$-vector
space of dimension $\dim A$.  
(For an account of the Lie algebra attached to a group
scheme over a field, see for example \cite{\MUMF, \S11}.)
The dimension of this vector space
is therefore a multiple of~$[E\:\Q]$, so that we have
$[E\:\Q] \mid \dim A$.  In particular, $[E\:\Q] \le \dim A$.

This observation motivates the study of abelian varieties $A/\Q$
whose endomorphism algebras contain number fields of maximal
dimension $\dim A$.  If $E$ is a number field of degree~$\dim A$
which is contained in~$\Q\otimes\End_\Q(A)$,
then the Tate modules $V_\ell(A)$ associated
with~$A$ are free of rank~two over $E\otimes_\Q\Qell$.  Accordingly,
the action of~$\GalQ$ on~$V_\ell(A)$ defines a representation
with values in~$\GL(2,E\otimes\Qell)$.  We
say that $A$ is of ``\GLTWO-type." 

Suppose that $B$ is of \GLTWO-type, and that
$F \subseteq \Q\otimes\End_\Q(B)$ is a number field of
degree $\dim B$.  Let $E$ be a number field containing~$F$,
and let $n=[E\:F]$.  After choosing a basis for $E$ over~$F$,
we find an embedding $E \subseteq M(n,F)$ of~$E$ into the
ring of $n$ by~$n$ matrices over~$F$.  Since $M(n,F)$ acts
naturally up to isogeny
on $A\seteq B\times\cdots\times B$ ($n$ factors), we obtain an
embedding $E\subseteq \Q\otimes\End_\Q(A)$.  Hence $A$ is again
of~\GLTWO-type. (We can summarize the situation by writing
$A=E\otimes_F B$.)
% In the category of abelian varieties up to isogeny over~$\Q$,
% $A$ represents the functor 
% $C \mapsto (\Q\otimes\Hom_\Q(C,B))\otimes_F E$.

We say that an abelian variety $A/\Q$ of \GLTWO-type is
{\it primitive\/} if it is not isogenous over~$\Q$ to 
an abelian variety
obtained by this matrix construction, relative to an extension
$E/F$ of degree~$n>1$ (cf.~\cite{\SHTAN, \S8.2}).



\proclaim{(2.1) Theorem}
Let $A$ be an abelian variety of
\GLTWO-type over~$\Q$.
Then the following conditions are equivalent: (i) $A$
is primitive; (ii) $A/\Q$ is simple; (iii) the endomorphism
algebra of~$A/\Q$ is
a number field whose degree
coincides with the dimension of~$A$.
\endproclaim
\startproof
Let $E$ be a number field of degree~$\dim A$ which is
contained in the $\Q$-algebra
$\Cal X\seteq\Q\otimes_\Z \End_\Q(A)$.
Let $D$ be the commutant of~$E$ in~$\Cal X$.
We claim that $D$ is a division algebra (cf.~\cite{\ANNALS,
Th.~2.3}).

     To prove the claim,
we must show that each non-zero
$\Q$-endomorphism of~$A$ which
commutes with~$E$ is an isogeny.
Let $\lambda$ be such an endomorphism and
let $B$ be the image
of~$\lambda$. Then $B$ is a non-zero abelian subvariety of~$A$.
The field $E$ operates on $B$ (up to isogeny), and thereby
acts by functoriality on the $\Q$-vector space $\Lie(B/\Q)$.
The dimension of $\Lie(B/\Q)$ is accordingly
a multiple of~$[E\:\Q]$; on the other hand, it
coincides with~$\dim B$.  Hence $B=A$, so that $\lambda$ is
an isogeny.

The Lie algebra $\Lie(A/\Q)$ may now be viewed as
a $D$-vector space. Because of this, the dimension of~$\Lie(A/\Q)$
is a multiple of 
$\dim_\Q(D)$.
In other words, we have $\dim(D) \mid \dim(E)$.
This gives the equality $D=E$; i.e., it shows that $E$ is its own
commutant in~$\Cal X$.

In particular, the center $F$ of~$\Cal X$
is a subfield
of~$E$.  We have then $\Cal X \approx M(n,Q)$,
where $Q$ is a division algebra with center~$F$.  If $Q$ has dimension
$t^2$ over~$F$, then $nt = [E\:F]$.  This follows from the fact
that $E$ is a maximal commutative semisimple subalgebra 
of~$\Cal X$,
which in turn follows from
the statement that $E$ is its own commutant in~$\Cal X$.

The structure of~$\Cal X$
shows that~$A$ is
isogenous (over~$\Q$) to a product of $n$ copies of an abelian
variety~$B$ whose algebra of $\Q$-endomorphisms is 
isomorphic to~$Q$.
The same Lie algebra argument we have already used shows
that $\dim_\Q(Q) \mid \dim(B)$, so that $n\cdot\dim_\Q(Q) \mid \dim(A)$.
This gives the divisibility $n t^2 [F\:\Q] \mid [E\:\Q]$, which
implies
$nt^2 \mid nt$.  One deduces that $t=1$, i.e., that
$Q=F$, and obtains the equality $n=[E\:F]$.  Hence the dimension of~$B$
coincides with the degree of $F$ over~$\Q$.  Also, we have
$\Q\otimes\End_\Q(A) \,\approx\, M(n,F)$.

In other words, $A$ is obtained from $B$ and~$F$ by the construction
we outlined above.  The equivalence of the three statements in
the theorem is now clear: each assertion is equivalent to the
equality $n=1$.
\endproof

\head 3. $\ell$-adic representations attached to primitive
abelian varieties over~$\Q$
of \GLTWO-type. \endhead\noindent
%
In what follows, we study primitive
abelian varieties of \GLTWO-type over~$\Q$.
Since we never encounter such abelian varieties which
are not primitive in the above sense, we will often
drop the word ``primitive" and refer simply to
abelian varieties over~$\Q$ of \GLTWO-type.

   To motivate the study of these varieties, we first
allude to the existence of a large
class of examples of (primitive) abelian varieties of
\GLTWO-type over~$\Q$.
Suppose that $f=\sum a_n q^n$ is
a normalized cuspidal
eigenform of weight two on a subgroup of
$\bold{SL}(2,\Z)$ of the form $\Gamma_1(N)$.
Then Shimura~\cite{\SHIM, Th.~7.14}
associates to~$f$ an abelian variety $A=A_f$
over~$\Q$ together with an action on~$A$ of the field
$E=\Q(\ldots,a_n,\ldots)$.  The variety $A_f$ may be
constructed as a quotient
of $J_1(N)$, the Jacobian of the standard
modular curve $X_1(N)$ \cite{\FACTOR}.
The dimension of~$A$ and the degree of~$E$ are equal.
It is well known (and easy to show)
that $E$ is the full algebra of endomorphisms
of~$A$ which are defined over~$\Q$ \cite{\MATHANN, Cor.~4.2}.
Thus, $A$ is of \GLTWO-type over~$\Q$.
For each $N\ge1$, the Jacobian $J_1(N)$
is isogenous to a product of abelian varieties of the
form~$A_f$ \cite{\MATHANN, Prop.~2.3}.

In \S4, we
study the converse problem: Suppose that
$A/\Q$ is of \GLTWO-type.
Is $A$ isogenous to 
a quotient of~$J_1(N)$, for some $N\ge1$?
We show that an affirmative answer is implied by
conjecture~(3.2.4$_?$) of Serre's article
\cite\DUKE\ on modular representations of~$\GalQ$.

We now begin the study of Galois representations 
attached to 
\GLTWO-type abelian varieties over~$\Q$.
Suppose that $A$ is such an abelian variety.
Let $E$ be the endomorphism algebra of~$A/\Q$.
Then $E$ is a number field which is
either a totally real number field
or a ``CM field," since each $\Q$-polarization of~$A$
defines a positive involution on~$E$.

Recall that
for each prime number $\ell$, the Tate module $V_\ell=V_\ell(A)$
is free of rank two over $E\otimes_\Q \Qell$.
For each prime $\lambda \mid\ell$ of~$E$, let $E_\lambda$
be the completion of~$E$ at~$\lambda$, and set
$V_\lambda\seteq V_\ell\otimes_{E\otimes\Qell} E_\lambda$.
Thus $V_\lambda$ is a two-dimensional vector space over~$E_\lambda$,
and $V_\ell$ is the direct sum of the~$V_\lambda$ with $\lambda\mid\ell$.
The action of~$\GalQ$ on~$V_\lambda$ defines a ``$\lambda$-adic
representation"~$\rho_\lambda$.  One knows that the collection
$(\rho_\lambda)$ (as $\lambda$ ranges over the set of finite
primes of~$E$) forms a strictly compatible system of $E$-rational
representations whose exceptional set is the set of prime
numbers at which $A$ has bad reduction.
(For background on this material,
see \cite{\DISCONT, \S11.10} and perhaps
\cite{\MYTHESIS, Ch.~II}.)
We will prove some facts about the $\lambda$-adic representations
$\rho_\lambda$ and their reductions mod~$\lambda$.


For each $\lambda$, let $\delta_\lambda\:\GalQ\to E^*_\lambda$
be the determinant of~$\rho_\lambda$.
The $\delta_\lambda$ form a compatible
system of $E$-rational one-dimensional representations
of~$\GalQ$.  In the case where $E$ is totally real, we have
$\det\rho_\lambda=\chi_\ell$, where $$\chi_\ell\:\GalQ\to\Zell^*$$
is the $\ell$-adic cyclotomic character, and where $\ell$ is the
prime of~$\Q$ lying below~$\lambda$ 
\cite{\MYTHESIS, Lemma~4.5.1}.  This formula must be modified
slightly in the case where $E$ is allowed to be a CM-field:


\proclaim{(3.1) Lemma}
There is a character of finite order $\epsilon\:\GalQ\to E^*$
such that $\delta_\lambda= \epsilon\chi_\ell$ for each finite
prime $\lambda$ of~$E$.  This character is unramified at
each prime which is a prime of good reduction for~$A$.
\endproclaim\startproof
Since the abelian
representations $\delta_\lambda$
arise from an abelian variety, they 
have the Hodge-Tate property.  It follows that they
are locally algebraic
in the sense of~\cite{\MG, Ch.~III};
see \cite{\MG, p.~III-50} and \cite{\MYTHESIS, Prop.~1.5.3}.
One deduces that the family $\delta_\lambda$ is
associated with an $E$-valued Grossencharacter of type~$A_o$
of the field~$\Q$ \cite{\MYTHESIS, p.~761}.  Since the
type-$A_o$ Grossencharacters of~$\Q$ are just products of Dirichlet
characters with powers of the ``norm" character,
there is an integer $n$ and an $E$-valued Dirichlet
character $\epsilon$ so that we have
$\delta_\lambda= \epsilon\chi_\ell^n$ for each $\lambda$.
Here, $\ell$ again denotes the residue characteristic of~$\lambda$.
(We will use the association $\ell\leftrightarrow\lambda$ from 
time to time without comment.)
We have blurred the distinction
between characters of $\GalQ$ of
finite order and Dirichlet characters: if $\epsilon$ is
a Dirichlet character, then its Galois-theoretic avatar
takes the value $\epsilon(p)$ on a Frobenius element for~$p$
in~$\GalQ$.

By the criterion of N\'eron-Ogg-Shafarevich, $\rho_\lambda$ is
ramified at a prime $p\ne\ell$ if and only if $A$ has bad
reduction at~$p$.  Thus, $\delta_\lambda$ is unramified at
a prime $p\ne\ell$ if $A$ has good reduction at~$p$.  In
other words, if $p$ is a prime of good reduction for~$A$,
and if $\ell\ne p$, then
$\epsilon\chi_\ell^n$ is unramified at~$p$.  Since $\chi_\ell$
is unramified at~$p$, $\epsilon$ is unramified at~$p$.

It remains to check that $n=1$.  For this, fix a prime number~$\ell$.
It is well known that the action of~$\GalQ$
on $\det_{\Qell}(V_\ell)$ is given by the character
$\chi_\ell^{\dim(A)} = \chi_\ell^{[E\:\Q]}$.
On the other hand, the
determinant of the action of~$\GalQ$ on~$V_\ell$ is
given by the character 
$$\prod_{\lambda \mid\ell} \Norm_{E_\lambda/\Qell}(\delta_\lambda)
= \Norm_{E/\Q}(\epsilon)\cdot\chi_\ell^{n\cdot [E\:\Q]}.$$
(Here, $\Norm$ denotes a norm.)
Since $\chi_\ell$ has infinite order, we deduce that
$\Norm(\epsilon)=1$ and that $n=1$.  \endproof


\proclaim{(3.2) Lemma}
Each character $\delta_\lambda$ is odd in the sense that it
takes the value $-1$ on complex conjugations
in~$\GalQ$.
\endproclaim\noindent
[Since $\chi_\ell$ is an odd character, the Lemma may be
reformulated as the statement that $\epsilon(-1)=+1$.]
For the proof,
we use the
comparison isomorphism
$$V_\lambda \,\approx\, H_1(A(\C),\Q)\otimes_E E_\lambda$$
resulting from an embedding $\Qbar \hookrightarrow \C$.
In this view of $V_\lambda$, the complex conjugation
of $\GalQ$ acts on $V_\lambda$ as $F_\infty\otimes 1$, where
$F_\infty$ is the standard ``real Frobenius" on~$H_1(A(\C),\Q)$
(cf.~\cite{\DELCORVALIS, \S0.2}).
In particular, $\delta_\lambda$ is odd if and only if we
have $\det F_\infty = -1$, where the determinant is taken
relative to the $E$-linear action of
$F_\infty$ on~$H_1(A(\C),\Q)$.

Since $F_\infty$ is an involution, and $H_1(A(\C),\Q)$ has
dimension~two, the indicated determinant is~$+1$ if and only
if $F_\infty$ acts as a scalar ($=\pm1$)
on~$H_1(A(\C),\Q)$.  To prove that $F_\infty$ does {\it not\/}
act as a scalar, we recall that $F_\infty\otimes 1$ permutes
the two subspaces $H_{0,1}$ and $H_{1,0}$
of~$H_1(A(\C),\Q)\otimes_\Q\C$
in the Hodge decomposition of~$H_1(A(\C),\Q)\otimes_\Q\C$.
\endproof

\proclaim{(3.3) Proposition}
For each $\lambda$, $\rho_\lambda$ is an absolutely irreducible
two dimensional representation of~$\GalQ$ over~$E_\lambda$.
We have $\End_{\Q_\ell[\GalQ]} V_\lambda = E_\lambda$.
\endproclaim\startproof
Faltings \cite\FALT\ proved that
$V_\ell$ is a
semisimple $\Qell[\GalQ]$-module whose commutant is
$E\otimes{\Qell}$.  (This is the Tate conjecture for endomorphisms
of~$A$.)  Since $V_\ell$ is the product of the~$V_\lambda$ with
$\lambda\mid\ell$, each module
$V_\lambda$ is semisimple over $\Qell[\GalQ]$
and satisfies $\End_{\Qell[\GalQ]}V_\lambda = E_\lambda$.
This implies that $V_\lambda$ is simple over~$E_\lambda$, and
that $\End_{E_\lambda} V_\lambda = E_\lambda$.
The absolute irreducibility follows.
\endproof

For each prime $p$ at which $A$ has good reduction, let
$a_p$ be the element of~$E$ such that
$$ a_p = \trace_{E_\lambda}(\Frob_p \mid V_\lambda) $$
whenever $\ell\ne p$.  Let $\overline{\phantom a}$ denote the canonical
involution on~$E$: this involution is the identity if $E$
is totally real, and the ``complex conjugation"
on~$E$ if $E$ is a CM~field.  The involution  $\overline{\phantom a}$
is the Rosati involution on~$E$ induced by every polarization of
$A/\Q$.

\proclaim{(3.4) Proposition}  We have $a_p = {\overline{a}}_p \epsilon(p)$
for each prime $p$ of good reduction.\endproclaim
\startproof
Let $\ell$ be a prime number. Let $\sigma\:E\hookrightarrow\Qellbar$
be an embedding of fields, and let $\sigmabar$ be the conjugate
embedding $x\mapsto \sigma(\overline{x})$.  Let $V_\sigma
= V_\ell \otimes_{E\otimes\Qell}\Qellbar$, where the tensor
product is taken relative to the map of $\Qell$-algebras
$E\otimes\Qell\to\Qellbar$ induced by~$\sigma$.
Define $V_\sigmabar$ similarly, using $\sigmabar$.

Fix a polarization of~$A$ defined over~$\Q$.  The associated
$e_{\ell^\nu}$-pairings of Weil induce a bilinear map
$\langle \, , \, \rangle\:V_\ell \times V_\ell \to \Qell(1)$,
where $\Qell(1)$ is (as usual) the $\Qell$-adic Tate
module attached to the $\ell$-power roots of unity in~$\Qbar$.
We have $\langle ex, y\rangle = \langle x, \overline e x\rangle$
for $e\in E$ and $x,y\in V_\ell$.  
Further, this pairing is $\GalQ$-equivariant in the sense that
we have
$\langle \ggg x, \ggg y\rangle = \ggg\langle x,y\rangle$
for $g\in\GalQ$.
After extending scalars from
$\Qell$ to $\Qellbar$, we find an isomorphism of
$\Qellbar[\GalQ]$-modules
$$ V_\sigmabar \approx \Hom(V_{\sigma},\Qellbar(1)).$$
Now (3.1) implies that
the determinant of $V_{\sigma}$ is a
one-dimensional $\Qellbar$-vector space
on which $\GalQ$ acts by the character $\ssigma\epsilon\chi_\ell$.  
Since $V_{\sigma}$ is of dimension two, this gives
$$\Hom(V_{\sigma},\Qellbar(\ssigma\epsilon\chi_\ell)) \approx
   V_{\sigma}.$$
In view of the fact
that $\Hom(V_{\sigma},\Qellbar(\ssigma\epsilon\chi_\ell))$
is the twist by~$\ssigma\epsilon$ of
$\Hom(V_{\sigma},\Qellbar(1))$,
we get $V_{\sigma}\approx V_\sigmabar(\ssigma\epsilon)$.
For $p\ne\ell$ a prime of good reduction, the
trace of $\Frob_p$ acting on $V_\sigma$ is $\sigma(a_p)$,
and similarly the trace of $\Frob_p$ acting
on~$V_\sigmabar(\ssigma\epsilon)$ is $\ssigma\epsilon(p)\sigmabar(a_p)
= \ssigma\epsilon(p)\sigma(\overline{a}_p)$.
This gives $a_p = \epsilon(p)\overline{a}_p$, as required.\endproof

\proclaim{(3.5) Proposition}
Let $S$ be a finite set of prime numbers including the set of primes
at which $A$ has bad reduction.  Then the
field $E$ is generated over~$\Q$ by the $a_p$ with $p\not\in S$.
\endproclaim
\startproof
The Proposition follows from
the Tate Conjecture for endomorphisms of~$A$
by a simple argument, which
we now sketch (see \cite{\MYTHESIS, pp.~788--789} for more
details).
Choose a prime number $\ell$, and observe that $\End_{\GalQ}V_\ell
= E\otimes\Qell$ because of Faltings's
results quoted in the proof of~(3.3).  Let $\overline{V}_\ell =
V_\ell\otimes\Qellbar$; then as a consequence we have
$\End_{\Qellbar[\GalQ]}\overline{V}_\ell =E\otimes\Qellbar$.
In addition, the semisimplicity of the $V_\ell$ implies that
$\overline{V}_\ell$ is semisimple as a $\Qellbar[\GalQ]$-module.
For each $\sigma\:E\hookrightarrow \Qellbar$, let $V_\sigma$
be as above.  Then the $V_\sigma$ are simple $\Qellbar[\GalQ]$-modules,
and they are pairwise non-isomorphic.  Indeed, they are
a priori semisimple, but the commutant of their product
is $\prod_\sigma \Qellbar$.  It follows that their traces
are pairwise distinct.  Since the trace of $\Frob_p$ acting
on $V_\sigma$ is $\sigma(a_p)$ (for $p\not\in S\cup \{\ell\}$),
the Cebotarev Density Theorem implies that the functions
$p\mapsto \sigma(a_p), p\not\in S\cup \{\ell\}$ are pairwise
distinct.
%
\endproof

For the next result, fix a finite set $S$ as in~(3.5), and let
$F$ be the subfield of~$E$ generated by the numbers $a_p^2/\epsilon(p)$
with $p\not\in S$.

\proclaim{(3.6) Proposition} The field $F$ is totally real.
The extension $E/F$ is abelian.\endproclaim\startproof
Let $\overline{\phantom a}$ again be the canonical complex
conjugation of~$E$.  For $p\not\in S$, we have
$${\overline{a}_p^2\over\overline{\epsilon}(p)} =
   {a_p^2\over \epsilon(p)^2}\epsilon(p) $$
by~(3.4).  The first assertion of the Proposition then 
follows.  For the second, let $t_p=a_p^2/\epsilon(p)$.
It is clear that $E$ is contained in the extension of~$F$
(in an algebraic closure of~$E$)
obtained by adjoining to~$F$ the square roots of all~$t_p$,
and all roots of unity.  This gives the second assertion.\endproof

We now consider the reductions of the $\lambda$-adic
representations~$\rho_\lambda$.  To do this directly,
replace $A$ by an abelian variety which is 
$\Q$-isogenous to~$A$ 
and which has the property that its ring of 
$\Q$-endomorphisms is the ring of integers $\Cal O$
of~$E$.
(This process does not change isomorphism classes of
the $\lambda$-adic representations $\rho_\lambda$.)
Write simply $A$ for this new abelian variety.
For each $\lambda$, consider the kernel $A[\lambda]$:
this is the group of $\Q$-valued points of~$A$ which are
killed by all elements of the maximal ideal $\lambda$ of~$\Cal O$.
The action of~$\Cal O$ on~$A[\lambda]$ makes $A[\lambda]$
into a two-dimensional vector space over the residue field
$\F_\lambda$ of~$\lambda$.  The $\F_\lambda$-linear action
of~$\GalQ$ on~$A[\lambda]$ defines a reduction
$\rholambdabar$ of~$\rho_\lambda$ (cf.~\cite{\MYTHESIS, II.2}).
(There should be no confusion with the involution $\overline{\phantom a}$
which appears above.)

\proclaim{(3.7) Lemma}
For all but finitely many $\lambda$, the representation
$\rholambdabar$ is absolutely
irreducible.
\endproclaim\noindent
%
A result of Faltings \cite{\FW, Theorem 1, page 204}
implies that
the following holds for almost all~$\lambda$:
$A[\lambda]$ is a
semisimple $\F_\ell[\GalQ]$ module whose commuting 
algebra is $\F_\lambda$.
The Lemma follows directly from this statement.\endproof
%
\head 4. Conjectural connection with modular forms.\endhead\noindent

We continue the discussion of~\S3, focusing
on the
possibility of linking $A$ with modular forms, at
least conjecturally.  According to~(3.7), $\rholambdabar$
is absolutely irreducible for almost all~$\lambda$.
For each $\lambda$ such that
$\rholambdabar$
is absolutely irreducible, conjectures
of Serre \cite{\DUKE, (3.2.3$_?$--3.2.4$_?$)} state that
$\rholambdabar$ is ``modular" in the sense that it 
arises from 
the space of mod~$\ell$ cusp forms of a specific
level $N_\lambda$, weight $k_\lambda$, and character
$\epsilon_\lambda$.

These invariants
are essentially constant as functions of~$\lambda$.  Rather than
study them for all $\lambda$, we will
restrict attention to those maximal ideals $\lambda$
which are odd,
prime to the conductor of~$A$,
are unramified in~$E$,
and have degree~one. 
Let $\Lambda$ be the set
of such ideals $\lambda$ with the property that
$\rholambdabar$ is absolutely irreducible.
%% What a mouthful!
Lemma 3.7 implies that $\Lambda$ is an infinite set.


\proclaim{(4.1) Lemma} The levels $N_\lambda$ are bounded as
$\lambda$ varies in~$\Lambda$.
\endproclaim\startproof
It follows from the definition of~$N_\lambda$ that this level
divides the conductor of the two-dimensional
$\ell$-adic representation $\rho_\lambda$.  
(To compare the two conductors, one can use the Hilbert Formula
of~\cite{\OGG, \S I}.)
The conductor of~$\rho_\lambda$
divides the conductor of the full $\ell$-adic
representation $V_\ell(A)$.  
According to results of A.~Grothendieck \cite{\SGA, Cor.~4.6},
this latter conductor is independent of~$\ell$.  (It is
by definition the conductor of~$A$.)\endproof

\proclaim{(4.2) Lemma} For all $\lambda\in\Lambda$, we
have $k_\lambda=2$.\endproclaim
\startproof
Take $\lambda\in\Lambda$.
The determinant of $\rholambdabar$ is the reduction
mod~$\lambda$ of $\delta_\lambda$.  By~(3.1), this
determinant is the product of
the mod~$\ell$ cyclotomic
character $\chiellbar$
and the reduction mod~$\lambda$ of~$\epsilon$.
The definition of~$\Lambda$ shows that $\ell$ is a prime
of good reduction for~$A$,
so that $\epsilon$ is unramified at~$\ell$ (Lemma~3.1).
Hence, if $I$ is an inertia group for~$\ell$ in~$\GalQ$,
we have $\det\rholambdabar\mid I = \chiellbar$.

Further, suppose that $D\subset\GalQ$ is a decomposition
group for~$\ell$.  It is clear that $\rholambdabar\mid D$
is finite at~$\ell$ \cite{\DUKE, p.~189}, since $A$ has good
reduction at~$\ell$.  Indeed,
the kernel of multiplication by~$\ell$
on~$A_{\Qell}$ extends to a finite flat group scheme $\Cal G$
over~$\Zell$
because of this good reduction \cite{\SGA, Cor.~2.2.9}.
The Zariski closure of~$A[\lambda]$
in~$\Cal G$ then prolongs $\rholambdabar$ to a group scheme of
type~$(\ell,\ell)$ over~$\Zell$.

By Proposition~4 of~\cite{\DUKE, \S2.8}, we find that $k_\lambda=2$.%
\endproof

\proclaim{(4.3) Lemma} For all but finitely
many $\lambda\in\Lambda$, we have $\epsilon_\lambda=\epsilon$.
\endproclaim
\noindent
One checks easily that $\epsilon_\lambda=\epsilon$ whenever~$\ell$
is prime to the order of~$\epsilon$.  We omit the details,
since Lemma~4.3 will not be used below.\endproof

\proclaim{(4.4) Theorem}
Let $A$ be an abelian variety over~$\Q$ of
\GLTWO-type.
Assume Serre's
conjecture \cite{\DUKE, (3.2.4$_?$)}
on representations of~$\GalQ$.
Then $A$ is isogenous to
a $\Q$-simple factor
of~$J_1(N)$, for some $N\ge1$.
\endproclaim
\startproof
Applying \cite{\DUKE, (3.2.4$_?$)} to 
the representations
$\rholambdabar$ with $\lambda\in\Lambda$, we find
that each $\rholambdabar$ 
arises from a newform of weight $k_\lambda=2$ and
level dividing $N_\lambda$.  
Since the $N_\lambda$'s
are bounded (Lemma~4.1),
 there are only a finite number of such
newforms.  

Hence there is a fixed newform $f=\sum a_nq^n$ which gives
rise to an infinite number of the $\rholambdabar$'s.
Explicitly, we have the following situation.  Let $R$ be
the ring of integers of the field $\Q(\ldots, a_n, \ldots)$.
For an infinite number of~$\lambda\in\Lambda$, there is
a ring homomorphism $\varphi_\lambda\: R \to \Flambdabar$
mapping $a_p$ to $\trace(\rholambdabar(\Frob_p))$
for all but finitely many primes~$p$.

Let $N$ be the level of~$f$, and let
$A_f$ be the quotient of~$J_1(N)$ which is 
associated to~$f$.
Let $\lambda$ be a prime for which there is a
$\varphi_\lambda$ as above.
By the Cebotarev Density Theorem, we have
$$A_f[\ell]\otimes_{R/\ell R}\Flambdabar
     \,\approx\, A[\lambda]\otimes_{\F_\lambda}\Flambdabar,$$
where $\Flambdabar$ is regarded as an $R/\ell R$-module
via~$\varphi_\lambda$.
Since $A_f[\ell]\otimes_{R/\ell R}\Flambdabar$ is a quotient
of $A_f[\ell]\otimes_\Fell\Flambdabar$, and since $\F_\lambda=\Fell$,
we have $$\Hom_{\Flambdabar[\GalQ]}(A_f[\ell]\otimes_\Fell\Flambdabar,
  A[\lambda]\otimes_\Fell\Flambdabar) \ne 0.$$
It follows that $\Hom_{\Fell[\GalQ]}(A_f[\ell], A[\lambda])\ne0$.

Hence we have $\Hom_{\Fell[\GalQ]}(A_f[\ell], A[\ell])\ne0$ for
an infinite number of prime numbers~$\ell$.
By the theorem of Faltings quoted above, we get
$\Hom_\Q(A_f,A)\ne0$.  Since $A$ is a simple abelian variety over~$\Q$,
$A$ must be a quotient of~$A_f$.  Since $A_f$ is, in turn, 
a quotient of~$J_1(N)$, we deduce that $A$ is a quotient of~$J_1(N)$.
\endproof

\head 5. Decomposition over~$\Qbar$.\endhead\noindent
Suppose again that $A/\Q$ is an abelian variety
of \GLTWO-type, and let $$E=\Q\otimes\End_\Q(A).$$
Let $\Cal X = \Q\otimes\End_{\Qbar}(A)$ be the algebra of
{\it all\/} endomorphisms of~$A$.


\proclaim{(5.1) Proposition {\rm [Shimura]}}
Suppose that $A_{\Qbar}$ has a non-zero abelian
subvariety of CM-type.
Then $A_{\Qbar}$ is isogenous to a power of a CM elliptic
curve.
\endproclaim
\noindent
This is Proposition~1.5 in \cite\CFHO.
For a generalization of this result, see
Proposition~5.2 of~\cite\HIDA.
See also the
discussion in~\S4 of~\cite\BONN.

\proclaim{(5.2) Proposition}
Suppose that $A_{\Qbar}$ has no non-zero abelian
subvariety of CM type.
Then the center of~$\Cal X$ is a subfield $F$ of~$E$.
%The field $F$ is totally real, and the extension $E/F$ is
%abelian.
The
algebra $\Cal X$ is isomorphic either to a matrix ring over~$F$,
or else to a ring of matrices over a % totally indefinite???
quaternion division algebra over~$F$.
\endproclaim
\startproof
%
We employ the same arguments used to prove~(2.1) above
and Theorem~2.3 of \cite\ANNALS.  Let $D$ be
the commutant of~$E$ in~$\Cal X$.  Clearly, $D$ is a division
algebra: otherwise, we can make $E$ act on a proper
non-zero
abelian subvariety of $A_{\Qbar}$, contrary to the
hypothesis that no abelian subvariety of~$A_{\Qbar}$ is
of CM~type. Also, $E$ is a subfield of~$D$; it is a maximal
commutative subfield because $A_{\Qbar}$ does not have
complex multiplication.  Hence $E$ is its own commutant in~$D$.
Since $D$ is the commutant of~$E$, we get $D=E$.  In particular,
the center of~$\Cal X$ is contained in~$E$, so that the
center is a subfield $F$ of~$E$.

Since $\Cal X$ is now a central simple algebra over~$F$, we have
$\Cal X \approx M(n,D)$, where $D$ is now
a division algebra with
center~$F$, and $n$ is a positive integer.  Suppose that $D$ has
dimension $t^2$ over~$F$; then $[E\:F]=nt$ since $E$ is a maximal
commutative subalgebra of~$\Cal X$.  Up to isogeny, $A_{\Qbar}$
is of the form $B^n$, where the endomorphism algebra of~$B$
contains $D$.  Following an idea of J.~Tunnell
(cf.~\cite{\DPP, Th.~1}), we
fix an embedding $\Qbar\hookrightarrow\C$ and
form the cohomology group $H^1(B(\C),\Q)$.  This is a $\Q$-vector
space of dimension 
$$ 2\dim(B) = {2\over n}\dim(A) = {2nt\over n}[F\:\Q]$$
with a functorial action of~$D$.  Hence the
$\Q$-dimension $t^2[F\:\Q]$ of~$D$ divides the dimension over~$\Q$
of~$H^1(B(\C),\Q)$, which is
$2t[F\:\Q]$. Thus $t\le 2$, so that either  $D=F$, or else
$D$ is a quaternion division algebra with center~$F$.
\endproof

As in Proposition~3.6, 
let $S$ be a finite set of primes containing the primes of
bad reduction for~$A$.  Let $F$ be the center of~$\Cal X$.
Then we have:

\proclaim{(5.3) Theorem}
The field $F$ is generated by the numbers $a_p^2/\epsilon(p)$
with $p\not\in S$.
\endproclaim
\noindent In other words, the field $F$ which appears in (3.6)
is the same as the field $F$ in~(5.2).

%To prove (5.3), we use the fact that, for each prime
%$\ell$,
%$F$ is the intersection
%of~$E$ with the center of $\Cal X\otimes_\Q\Qellbar$.
To prove (5.3), we
let $\ell$ be a prime which
splits completely in~$E$, so that all
embeddings $E\hookrightarrow\Qellbar$ take values in~$\Qell$.
Choose a finite extension $K$ of~$\Q$ such that all endomorphisms
of~$A$ are defined over~$K$, and let $H$ be the
corresponding open subgroup of~$\GalQ$.
Replacing $H$ be a smaller subgroup of~$\GalQ$ if necessary,
we may assume that $H$ is contained in the kernel of~$\epsilon$.
We have $\Cal X\otimes\Qell = \End_{\Qell[H]}{V}_\ell$, by
Faltings's results \cite{\FALT}.
The center of~$\Cal X\otimes\Qell$ is $F\otimes\Qell$.

The Tate module ${V}_\ell$ decomposes as
a product $\prod_\sigma V_\sigma$, where
$\sigma$ runs over the set $\Sigma$ of
embeddings $\sigma\:E\to \Qell$,
and where $V_\sigma=V_\ell\otimes_{E\otimes\Qell}\Qell$,
with $\Qell$ being regarded as an $E\otimes\Qell$ module
via~$\sigma$.
(Cf.~the proof of~(3.4).) 
Each
$V_\sigma$ is a simple $\Qell[H]$-module because $A$ has
no CM subvariety and because the action of~$H$ on
$V_\ell$ is semisimple (Faltings).  Hence $\End_H V_\sigma =\Qell$
for each~$\sigma$.

For each prime $v$ of~$K$ which is prime to $\ell$ and the set
of bad primes for~$A$, there is a ``trace of Frobenius" $t_v\in E$
associated with~$v$.  We have $\trace(\Frob_v\mid V_\sigma) =
\sigma(t_v)$ for each~$\sigma$.  One knows for $\sigma,\tau\in\Sigma$
that $V_\sigma$ and $V_\tau$ are isomorphic $\Qell[H]$-modules
if and only if $\sigma(t_v)=\tau(t_v)$ for all~$v$, i.e., if
and only if $\sigma|_L = \tau|_L$, where $L=\Q(\ldots,t_v,\ldots)$
(cf.~\cite{\MYTHESIS, \S IV.4}).
This implies that the center of~$\Cal X\otimes\Qell$ is~$L\otimes\Qell$.
Thus $F\otimes\Qell=L\otimes\Qell$
(equality inside $E\otimes\Qell$), which implies that~$F=L$.  

Suppose now that $\sigma=\tau$ on~$L$, so that $V_\sigma\approx
V_\tau$ as representations of~$H$.
A simple argument shows that 
there is a
character $\phi\:\GalQ\to\Q^*_\ell$ such that $V_\sigma \approx
V_\tau\otimes\phi$ as $\Qell[\GalQ]$-modules.
The character $\phi$
is necessarily
unramified at all primes $p\ne\ell$ which are
primes of
good reduction for~$A$.
Taking traces, we get 
$\sigma(a_p) = \phi(p)\tau(a_p)$ for all such primes.
By considering determinants, we get
$\ssigma\epsilon= \phi^2\cdot\ttau\epsilon$.
These two equations show that $\sigma$ and~$\tau$ agree on
$a_p^2/\epsilon(p)$ for all good primes $p\ne\ell$.

It follows by Galois theory that we have $a_p^2/\epsilon(p)\in L$
for all good reduction primes $p\ne\ell$.  Hence $a_p^2/\epsilon(p)\in F$
for all such~$p$.  By varying~$\ell$ we obtain the inclusion
$\Q(\ldots,a_p^2/\epsilon(p),\ldots) \subseteq F$, where $p$ runs
over the set of all primes of good reduction for~$A$.

   To prove the opposite inclusion, we must show that if
$\sigma(a_p^2/\epsilon(p)) = \tau(a_p^2/\epsilon(p))$ for all~$p$,
then $V_\sigma$ and $V_\tau$ are $H$-isomorphic.
By the Cebotarev Density Theorem, the hypothesis implies that 
the functions on~$\GalQ$ ``$\trace^2/\det$" are the same for
$V_\sigma$ and~$V_\tau$.  In particular, we have
$\trace(h| V_\sigma) = \pm \trace(h | V_\tau)$ for
all $h\in H$.  This equality implies that $V_\sigma$ and $V_\tau$
become isomorphic after $H$ is replaced by an open subgroup $H_o$
of~$H$ (cf.~\cite{\SERREINV, p.~324}).  Since $H$ was already
chosen ``sufficiently small," we find that $V_\sigma$
and $V_\tau$ are indeed isomorphic as $\Qell[H]$-modules.\endproof

\proclaim{(5.4) Corollary} The center $F$ of~$\Cal X$ is a
totally real number field.  The extension $E/F$ is abelian.
\endproclaim\startproof
The Corollary follows from (5.3) and~(3.6).
\endproof

Let $g$ be an element of~$\GalQ$, and consider the automorphism
$x\mapsto \ggg x$
of~$\Cal X$ which is induced by~$g$.  
This automorphism is necessarily inner (Skolem-Noether theorem),
and it fixes $E$; therefore it is given by conjugation by an
element $\alpha(g)$
of~$E^*$ which is well defined modulo $F^*$.  The map
$\alpha\:\GalQ\to E^*/F^*$ is a continuous homomorphism
(cf.~\cite{\DPP, p.~268}).  It is a fact that $\alpha$ is
unramified at all primes $p$ at which $A$ has semistable reduction
\cite{\ANNALS, Th.~1.1}.

\proclaim{(5.5) Theorem}
We have $\alpha^2\equiv\epsilon$ {\rm (mod~$F^*$)}. Moreover,
suppose that $p$ is a prime of good reduction for~$A$
such that $a_p\ne0$.  Then $\alpha(\Frob_p)\equiv a_p$
{\rm (mod~$F^*$)}.
\endproclaim
\startproof To prove
the first assertion, let $\ell$ be a prime which
splits completely in~$E$.  We must prove that
$\sigma(\alpha^2(g)/\epsilon(g))=
\tau(\alpha^2(g)/\epsilon(g))$ whenever $\sigma$ and~$\tau$ are embeddings
$E\hookrightarrow \Qell$ which agree on~$F$.

If $\sigma$ and~$\tau$ have this property, then there is
a character $\phi\:\GalQ\to\Q^*_\ell$ for which $V_\sigma\approx
V_\tau\otimes\phi$ (the notation is as in the proof of~(5.3)).
The Galois group $\GalQ$ acts on $\Hom(V_\sigma,V_\tau)$
by multiplication by~$\phi(g)^{-1}$, but also by conjugation by~$\alpha(g)$.
Since $\alpha(g)$ acts on~$V_\sigma$ and $V_\tau$ by
$\ssigma\alpha(g)$ and $\ttau\alpha(g)$ (respectively),
$\alpha(g)$ acts on $\Hom(V_\sigma,V_\tau)$ by 
$\ttau\alpha(g)/\ssigma\alpha(g)$.
Hence, $\phi(g)=\ssigma\alpha(g)/\ttau\alpha(g)$
in~$\Qell$.
On the other hand, as remarked during the proof of~(5.3),
we have $\ssigma\epsilon=\phi^2\cdot\ttau\epsilon$.
The two equalities give the required conclusion
$$ \sigma(\alpha^2/\epsilon) = \tau(\alpha^2/\epsilon).$$

For the second assertion, we choose $\ell\ne p$.
With the notation as above,
$\phi(p) = \ssigma\alpha(\Frob_p)/\ttau\alpha(\Frob_p)$.
However, we have $\sigma(a_p) = \phi(p)\tau(a_p)$, as
noted during the proof of~(5.3). On comparing two
formulas for~$\phi(g)$, we get $\sigma(\alpha(\Frob_p)/a_p)=
\tau(\alpha(\Frob_p)/a_p)$.\endproof

\proclaim{Remark}\rm
It should be easy to prove
that the set of~$p$ for which $a_p=0$ has
density~0, since the image of~$\GalQ$ in $\Aut V_\sigma$
is open in $\Aut V_\sigma$, for any~$\sigma$.
\endproclaim
\noindent
Let $\tilde\alpha$ be a lift of~$\alpha$ to a function
$\GalQ\to E^*$. 
The function
$$c\:(g_1,g_2)\mapsto {\tilde\alpha(g_1)\tilde\alpha(g_2)
 \over\tilde\alpha(g_1g_2)}$$
is then a 2-cocycle on~$\GalQ$ with values in~$F^*$.  (We regard
$F^*$ as a $\GalQ$-module with trivial action.)
The image $[c]$ of~$c$ in $H^2(\GalQ,F^*)$ is independent
of the choice of~$\tilde\alpha$.  Since $\alpha^2$ lifts to the
character $\epsilon$, it is clear that $[c]$
has order dividing~two.

Consider the map
$$ \xi\:H^2(\GalQ,F^*) \, \longrightarrow \, H^2(\Gal(\Qbar/F),\Qbar^*)
           = \Br F$$
obtained from the restriction
map $H^2(\GalQ,F^*)\to H^2(\Gal(\Qbar/F),F^*)$ 
and the map $H^2(\Gal(\Qbar/F),F^*) \to H^2(\Gal(\Qbar/F),\Qbar^*)$
induced by the inclusion of $F^*$ into $\Qbar^*$.
(We view $\Qbar^*$ as a $\Gal(\Qbar/F)$-module in the
standard way; i.e., we use the Galois action.  The notation
``$\Br F$" indicates the Brauer group of~$F$.)

\proclaim{(5.6) Theorem {\rm [Chi]}}
The class of~$\Cal X$ in~$\Br F$ coincides with $\xi([c])$.
\endproclaim
\startproof
According to \cite{\CHI, Th.~3.4}, the algebra $\Cal X$
is isomorphic to a twisted matrix algebra $(\End_F E)(\alpha)$.
Theorem~4.8 of~\cite\CHI\ expresses $[(\End_F E)(\alpha)]\in\Br F$
as the image of a certain two-cocycle whose values are Jacobi
sums.
By \cite{\DPP, Prop.~1}, the class of this Jacobi-sum
cocycle coincides with the class of the two-cocycle
%
$$ (g,h)\mapsto \tilde\alpha(h)^{g-1}
  = {\ggg\tilde\alpha(h)\over\tilde\alpha(h)}$$
%
on $\Gal(\Qbar/F)$.  We wish to compare this with the two-cocycle
$$ (g,h)\mapsto
     {\tilde\alpha(g)\tilde\alpha(h)\over\tilde\alpha(gh)}.$$
The product of the two is the map
$$ (g,h)\mapsto
    {\ggg\tilde\alpha(h)\tilde\alpha(g)\over\tilde\alpha(gh)},$$
which is a coboundary.  Hence the class of~$\Cal X$ in~$\Br F$
is the negative of $\xi([c])$.  Since $[c]$ (and
$[\Cal X]$) have order two, we
get the Theorem as stated.\endproof

A final remark about $\Cal X$ concerns the isogeny class
of the simple factors of~$A_{\Qbar}$.  We have $\Cal X\approx
M(n,D)$ for some positive integer~$n$, where $D$ is a
division algebra of dimension one or~four over~$F$.
Accordingly, $A_{\Qbar}$ decomposes up to isogeny as
the product of $n$ copies a
simple abelian variety $B$ over~$\Qbar$ 
whose endomorphism algebra is isomorphic to~$D$.
Since $A$ is defined over~$\Q$, we have
$$ \ggg B\times\cdots\times\ggg B \,\,\sim \,\,\ggg A_{\Qbar}
 \,\,\sim A_{\Qbar} \,\,\sim B\times\cdots\times B$$ for each $g\in\GalQ$.
(The sign ``$\sim$" indicates an isogeny.)
By the uniqueness of decomposition up to isogeny, we have
$\ggg B\sim B$.  One says that the isogeny class of~$B$
is ``defined over~$\Q$."

Consider the special case where $n=\dim A$, so that $B$ is
of dimension~one.  The elliptic curve $B$ is then 
isogenous to
each of its conjugates (over~$\Qbar$).  Borrowing 
(and bending) a
term used by B.~H.~Gross \cite\BHG, we say that
$B$ is a ``$\Q$-curve."

\head 6. $\Q$-curves as factors of abelian varieties of \GLTWO-type.%
\endhead\noindent
We have just proved:
Suppose that $C$ is an elliptic curve over~$\Qbar$ which occurs
as a simple factor of a primitive abelian variety of \GLTWO-type
over~$\Q$ with no complex multiplication over~$\Qbar$.
Then $C$ is a $\Q$-curve; it is, more
precisely, a $\Q$-curve with no complex multiplication.

In this \S, we prove the converse:

\proclaim{(6.1) Theorem}
Suppose that $C$ is an elliptic curve over~$\Qbar$ with no
complex multiplication.  Assume that $C$ is isogenous to each
of its conjugates $\ggg C$ with $g\in\GalQ$.
Then there is a primitive abelian variety $A$
of \GLTWO-type over~$\Q$ such that $C$ is a simple factor
of~$A$ over~$\Qbar$.
\endproclaim

\proclaim{(6.2) Corollary}
Suppose that $C$ is as in~(6.1).
Assume Serre's
conjecture \cite{\DUKE, (3.2.4$_?$)}
on representations of~$\GalQ$.  
Then $C$ is a simple factor, over~$\Qbar$, of $J_1(N)$ for
some $N\ge1$.
\endproclaim
\noindent
The Corollary follows directly from (6.1) and~(4.4).\endproof
\medskip
\noindent
{\sl Proof of~(6.1).}\enspace
Let $C$ be as in the statement of the Theorem.  We can find
a model $C_o$
of~$C$ over a number field $K\subset\Qbar$.  
We may assume that $K/\Q$ is a Galois extension.
For each $g\in\Gal(K/\Q)$, $C_o$ and $\ggg C_o$ are
$\Qbar$-isogenous.  Enlarging $K$ if necessary, we may assume
that there are isogenies $\mu_g\: \ggg C_o\to C_o$ defined
over~$K$.

The map
$$ c\:(g,h)\mapsto \mu_g \ggg \mu_h \mu^{-1}_{gh}$$
may be regarded as $\Q^*$-valued, since $\Q\otimes\End_K(C_o) =\Q$.
A short
computation shows that $c$ is a two-cocycle on~$\Gal(K/\Q)$
with values in~$\Q^*$.  The class of~$c$ in $H^2(\Gal(K/\Q),\Q^*)$
is independent of the choices of the $\mu_g$.
By inflation, we may (and will) regard $c$ as a locally
constant function on~$\GalQ$.

\proclaim{(6.3) Theorem {\rm [Tate]}}
Let $M$ be the discrete $\GalQ$-module $\Qbar^*$ with
trivial action.  Then $H^2(\GalQ,M)=0$.
\endproclaim\noindent
This theorem of Tate is proved as Theorem~4
in Serre's article \cite\JPSDURHAM,
with $\Qbar$ replaced by~$\C$.  
The proof exposed by~Serre 
begins with the observation that $$H^2(\GalQ,\C^*)=H^2(\GalQ,W),$$
where $W$ is the torsion subgroup of~$\C^*$, i.e., the group
of complex roots of unity.  This observation follows from
the fact that $\C^*/W$ is uniquely divisible.  
Since $\GalQ$ acts trivially on~$\C^*$, $W$ is $\GalQ$-isomorphic
to~$\Q/\Z$.  One proves that $H^2(\GalQ,\Q/\Z)=0$; in fact, we
have $H^2(\Gal(\overline{K}/K),\Q/\Z)=0$ whenever $K$ is a
local or global field \cite{\JPSDURHAM, \S6.5}.

Since the quotient of~$M$ by its torsion subgroup is 
uniquely divisible, and since the torsion subgroup of~$M$
is isomorphic to~$\Q/\Z$, we obtain $H^2(\GalQ,M)=H^2(\GalQ,\Q/\Z)=0$.\endproof

%what would happen over a finite field?

Because of Tate's theorem, there is a locally constant
function $\alpha\:\GalQ\to\Qbar^*$ such that 
we have the identity among functions $G\times G\to\Qbar^*$
$$ c(g,h)= {\alpha(g)\alpha(h)\over\alpha(gh)}.$$
After again enlarging $K$, we may identify $\alpha$
with a function on~$\Gal(K/\Q)$ and regard $g$ and~$h$
as elements of this Galois group.

Let $E$ be the extension of~$\Q$ generated by the values
of~$\alpha$.  The definition of~$c$ shows that we have
$$ c(g,h)^2 = {\deg\mu_g \deg\mu_h\over\deg\mu_{gh}},$$
where ``deg" denotes the degree of an isogeny between
elliptic curves (or, more generally, of a non-zero element
of $\Q\otimes\Hom(C_1,C_2)$, where $C_1$ and $C_2$ are
elliptic curves).  It follows that the function
$$ \epsilon\:g\mapsto {\alpha^2(g)\over\deg\mu_g}$$
is a Dirichlet character $\Gal(K/\Q)\to E^*$.
Because $\alpha^2 \equiv \epsilon$ (mod~$\Q^*$), the
field $E$ is an abelian extension of~$\Q$ (cf.~Prop.~3.6).

If one performs the analysis of~\S3 on the abelian
variety~$A$ which we construct below, one finds (e.g., in~(3.1))
another Dirichlet character called $\epsilon$. 
It is very likely that the two~$\epsilon$'s are equal
up to ``sign," i.e., possible inversion (cf.~Lemma~7.1 below).
% A comparison
% between the two $\epsilon$'s will presumably appear in
% a later version of this article.


   To simplify notation, let us write simply $C$ for the elliptic
curve $C_o$ over~$K$ and $C_{\Qbar}$ for the curve originally
called~$C$.  Let $B$ be the abelian variety $\Res_{K/\Q} C$,
where ``Res" is Weil's ``restriction of scalars"
functor~\cite{\WEIL, \MILNE}.
Then $B$ is an abelian variety over~$\Q$ of dimension $[K:\Q]$.
It represents the functor on~$\Q$-schemes
$S\mapsto C(S_{K})$; in particular, we have
$\Hom_\Q(X,B) = \Hom_K(X_{K},C)$ whenever $X$ is an abelian
variety over~$\Q$.

Applying this formula in the special case where $X=B$,
we find $\End_\Q(B)=\Hom_K(B_{K},C)$.
On the other hand 
\cite{\WEIL, p.~5},
$$B_{K} = \prod_{\sigma\in\Gal(K/\Q)} \ssigma C.$$
Hence we have
$$ \Q\otimes\End_Q(B) = \prod_\sigma \Q\otimes\Hom_K(\ssigma C,C).$$
Since $\Q\otimes\Hom(\ssigma C,C)$ is the one-dimensional
vector space generated by $\mu_\sigma$, we find that
$\Cal R\seteq\Q\otimes\End_\Q(B)$
may be written as $\prod \Q\cdot\mu_\sigma$.
Let $\lambda_\sigma$ be the element of $\Cal R$ which 
corresponds to
$\mu_\sigma\:\ssigma C\to C$; then $\Cal R$ is a $\Q$-algebra
with vector space basis $\lambda_\sigma$.

\proclaim{(6.4) Lemma} We have $\lambda_\sigma\lambda_\tau
= c(\sigma,\tau)\lambda_{\sigma\tau}$ in~$\Cal R$ for
$\sigma,\tau\in\Gal(K/\Q)$.
\endproclaim
\startproof
Each map $\lambda_\sigma$ acts on $B_{K}=\prod_g \ggg C$ as a
``matrix": it sends the factor
${}^{g\sigma}\mkern-\thinmuskip C$ to~$\ggg C$
by $\ggg\mu_\sigma$ (cf.~\cite{\BHG. \S15}).
A short computation using the identity
$$ \mu_\sigma\ssigma\mu_\tau = c(\sigma,\tau)\mu_{\sigma\tau}$$
gives the desired formula.
\endproof

The algebra $\Cal R$ is thus a ``twisted group algebra"
$\Q[\Gal(K/\Q)]$ in which we have the multiplication table
$[\sigma][\tau]=c(\sigma,\tau)[\sigma\tau]$.
The obvious map of $\Q$-vector spaces
$$\omega\:\Cal R\to E, \qquad \lambda_\sigma\mapsto \alpha(\sigma)$$
is in fact a surjective homomorphism of $\Q$-algebras
because of~(6.4).

In \cite\GRBU, $\Q\otimes\End_\Q(B)$ is studied in an analogous
situation where $C$ has complex multiplication, and where
$K$ is the
Hilbert class field of the field of complex multiplication.
(The curve $C$ is then a ``$\Q$-curve" in the original sense
of the term.)  Criteria are given for $\Q\otimes\End_\Q(B)$
to be (i) a commutative algebra, i.e., a
product of number fields, and (ii) a product of
{\it totally real\/} fields.  It might be interesting to formulate
similar criteria in our context.

Let $T$ be the abelian variety $\prod_\sigma C$ over~$K$.  We
write $C_\sigma$ for the copy of~$C$ in the $\sigma$th place,
so that $T$ becomes the product $\prod C_\sigma$.
We have a map $\Cal R\to \Q\otimes\End T$
given as follows: For $g\in\Gal(K/\Q)$, the element
$\lambda_g = [g]$ of~$\Cal R$ acts on~$T$ by sending 
$C_\sigma$ to~$C_{g\sigma}$
by the map (of elliptic curves up to isogeny)
``multiplication by~$c(g,\sigma)$."  One checks directly that
this is a homomorphism of $\Q$-algebras.  The variety~$T$ will
be interpreted as $\Cal R\otimes_\Q C$ by readers who are
fond of such tensor-product constructions.

Let $\iota\:T\,\buildrel\sim\over\to\,
     B_{K} = \prod \ssigma C$ be the isomorphism
of abelian varieties up to isogeny which takes the factor
$C_\sigma$ of~$T$ to the factor ${}^{\sigma^{-1}}\mkern-\thinmuskip
C$ of~$B_{K}$, via
the map ${}^{\sigma^{-1}}\mkern-\thinmuskip\mu_\sigma$.


\proclaim{(6.5) Proposition} The map $\iota$ is $\Cal R$-equivariant,
for the action of~$\Cal R$ on~$T$ just defined and for the
structural action of~$\Cal R$ on~$B$.
\endproclaim\startproof
The proof of this Proposition is an uninteresting computation,
which is omitted.


\proclaim{(6.6) Corollary} The Lie algebra $\Lie(B/\Q)$ is
a free $\Cal R$-module of rank~one.\endproclaim
\startproof
The statement to be proved is true if and only if $\Lie(B_{K}/K)$
is free of rank~one over $R\otimes_\Q K$.
By~(6.5), $\Lie(B_{K}/K)$ may be identified with $\Cal R\otimes_\Q
\Lie(C/K)$, with $\Cal R$ operating trivially on the second factor.
Hence $\Lie(B_{K}/K)$ is indeed free of rank~one over 
$\Cal R\otimes_\Q K$.
\endproof

%can we prove that the l-adic representations are free of rank 2 over
% R\otimes\Ql?  If so, what is the "determinant"?  This has a sense
% only for the abelian part of \Cal R!

   To complete the proof of~(6.1), we let $A$ be the abelian
variety $E\otimes_{\Cal R} B$, where $E$ is viewed as a
$\Cal R$-module via~$\omega$.  Explicitly,
use the fact that $\Cal R$ is a semisimple 
$\Q$-algebra to write 
$\Cal R$ as a direct sum of its quotient $E$ with
the kernel of the map $\omega$.  Let $\pi\in\Cal R$ be
the projector onto~$E$, and let $A\subseteq B$ be the image
of~$\pi$, viewed as an endomorphism of~$A$ up to isogeny.
(In other words, $A$ is the image of $m\cdot\pi$, where
$m$ is a positive integer chosen so that $m\cdot\pi$ is
a true endomorphism of~$B$.)
Then $A$ is an abelian subvariety of~$B$, defined over~$\Q$,
whose algebra of $\Q$-endomorphisms is~$E$.
By~(6.6), $E$ acts without multiplicity on~$\Lie(B)$ and
therefore, in particular, without multiplicity on~$\Lie(A)$.
Hence $A$ has dimension equal to~$[E\:\Q]$, which means
that $A$ is of~\GLTWO-type.  (It is clear that $A$ is non-zero
because $\pi$ is non-zero.)

Since $B_K$ is isogenous to a product of copies of~$C$,
the same holds true for~$A_K$.  Thus $C$ is 
a quotient of~$B_K$.
\endproof

% what happens to the determinant of the l-adic reps?  Is det rho_l
% equal to chi_l times epsilon, where epsilon is the character
% with values in E that is attached to alpha?

\head 7. $\Q$-curves over quadratic fields.%
\endhead\noindent


Suppose that $C$ is a $\Q$-curve as above and that 
$K$ is a
quadratic field.  Let $\sigma$ be the non-trivial
automorphism of $K$ over~$\Q$.  
Then, by hypothesis, there is a $K$-isogeny
$\mu=\mu_\sigma\:\ssigma C\to C$.
We take the identity map
for $\mu_1$, where ``1" is the identity automorphism of $K$.
The cocycle~$c$ takes the value~1 on all elements of
$\Gal(K/\Q)\times\Gal(K/\Q)$ other than $(\sigma,\sigma)$.
Its value on that pair is the non-zero
integer $m$ such that
$\mu\circ\ssigma\mu$ is multiplication by $m$ on~$C$.
The algebra~$\Cal R$ may be written $\Q[X]/(X^2-m)$, where
$X$ corresponds to the element of~$\Cal R$ we have
been calling~$[\sigma]$.

Let us split $c$ by defining $\alpha\:\Gal(K/\Q)\to\Qbar^*$
to be the map taking $1$ to~$1$ and $\sigma$ to a square
root of~$m$.  The character
$$\theta\: g\mapsto {\alpha^2(g)\over\deg\mu_g}$$
is then trivial if $m$ is positive and of order two
if $m$ is negative.  In the case where $\theta$ is of order
two, it is an isomorphism $\Gal(K/\Q)\buildrel\sim\over\to \{ \pm 1\}$.


If $m$ is a perfect square, then we have $E=\Q$ in the
notation of~\S6.  The abelian variety $A$ is then
a model of~$C$ over~$\Q$.
\medskip

{\it Assume for the rest of this \S\ that $m$ is not a
perfect square.}  Then $\Cal R=E$ is a quadratic number field,
and we have $B=A$ in the notation of~\S6.
The field $E$ is real if $m$ is positive and imaginary
if $m$ is negative.  Thus $E$ is real if and only if $\theta$
is trivial.

The $\lambda$-adic representations of~$A$
define a Dirichlet character $\epsilon$ (Lemma~3.1).
According to~(3.2), $\epsilon$ is an even
character. Also, $\epsilon$ is non-trivial if and only
if $E$ is imaginary~(3.4).  Thus $\epsilon$ if non-trivial
if and only if $\theta$ is non-trivial.

\proclaim{(7.1) Lemma}
The characters $\epsilon$ and $\theta$ are equal.\endproclaim\startproof
% We have already seen that $\epsilon$ is non-trivial if
% and only if $\theta$ is non-trivial.
By~(6.5), $A_K$ is $K$-isogenous to the abelian variety
``$E\otimes C$," i.e., to the product of two copies of~$C$ with 
$E$ acting through a regular representation $E\hookrightarrow M(2,\Q)$.
In particular, the $\lambda$-adic representations of~$A_K$ are just
the $\ell$-adic representations of~$C$, viewed as taking values in
$\bold{GL}(2,E_\lambda)$ rather than in~$\bold{GL}(2,\Qell)$.
This implies that the determinants of the $\rho_\lambda |_{\Gal(\Qbar/K)}$
are the cyclotomic characters~$\chi_\ell$.
Hence $\epsilon$ is trivial on~$\Gal(\Qbar/K)$, and therefore
may be identified with a character of the
group~$\Gal(K/\Q)$, whose order is~two.
Since $\theta$ is also a character of this latter group, and since
the two characters are simultaneously non-trivial, they are
equal.\endproof




As mentioned above,
it seems very likely that Lemma~7.1 
(quite possibly in the form $\epsilon=\theta^{-1}$)
generalizes to the situation
of~\S6.

\proclaim{(7.2) Proposition {\rm [Serre]}}
At least one of the two quadratic fields $E$, $K$ is a
real quadratic field.\endproclaim  
%If $E$ is an imaginary quadratic field,
%then $K$ is a real quadratic field.
%\endproclaim
\startproof
We give two proofs, the first of which was communicated to the
author by Serre:
Assume that $K$ is a complex quadratic field.
After we
embed $K$
in~$\C$, the automorphism $\sigma$ of~$K$
becomes the restriction to~$K$ of $\bar{\phantom L}$,
complex conjugation on~$\C$.
Choose a holomorphic differential  $\omega$
on~$C$.
Then
$C_\C$ may be identified with
the curve $\C/L$, with $L\subset\C$ the period lattice
of~$\omega$.  The curve $\ssigma C$ becomes the complex
conjugate $\C/\bar L$
of~$C$.  The isogeny
$\mu$ is induced by the map ``multiplication by $\gamma$"
on~$\C$, for some non-zero complex number~$\gamma$.
The integer $m$ may be identified with
$\gamma\bar\gamma$, and is therefore positive.
This means that $E$ is a real quadratic field.

Another proof can be given as follows.  Assume that $E$ is
an imaginary quadratic field.  Then $\epsilon$ is a
non-trivial character of~$\Gal(K/\Q)$.  This character
is {\it even\/} by~(3.2).
Therefore $K$ is real.
\endproof

In connection with Proposition~7.2, it is natural to ask whether
each of the three possibilities allowed by the Proposition do,
in fact, occur.
The case where $E$ is imaginary, so that $K$ is real,
was treated in detail
by Shimura~\cite{\CFHO, \S10}, who constructed a family of
numerical examples.
This case was later studied by Serre \cite{\DUKE, p.~208},
who pointed out that Theorem~4.4 holds in this context.
A situation where $E$ is a real quadratic field and $K$
is an imaginary quadratic field is described
by~Koike in~\cite{\KOIKE, \S1}.
In this example, 
the abelian variety $A$ is associated to a weight-two
newform on~$\Gamma_o(81)$.
One has
$E=\Q(\sqrt3)$ and $K=\Q(\sqrt{-3})$.


To 
exhibit an example where both $E$ and~$K$
are real, we consider the eight-dimensional
complex vector space $S$
of weight-two cusp forms on~$\Gamma_o(169)$.
According to~\cite{\ANTIV, Table~5}, 
one can find a (normalized) newform $f=\sum a_n q^n\in S$ whose coefficients
generate a quadratic field $E$.  Moreover, the only two such forms
are $f$ and $\ssigma f = \sum \ssigma a_n q^n$, where $\sigma$ is
the non-trivial automorphism of $E$ over~$\Q$.
Since $f$ is a form with trivial ``Nebentypus" character, $E$ is
a priori a real quadratic field.  In fact, the author has
learned from~D.~Zagier that tables
constructed by Cohen and Skoruppa show that $E=\Q(\sqrt 3)$.

Let $\phi$ be the quadratic Dirichlet character with
conductor~13.  Then $f\otimes\phi\seteq \sum \phi(n)a_n q^n$
is again a normalized newform in~$S$; its coefficients 
generate the same quadratic field $E$ as the coefficients of~$f$.
Hence we have either $f=\phi\otimes f$, or else $\ssigma f=\phi\otimes
f$.  The former possibility is excluded by~\cite{\BONN, Th.~4.5},
since $\phi$ is an even character (i.e., since $\phi$ corresponds
to a real quadratic field).  Hence $f$ is a form with an
``extra twist" by~$\phi$: we have $\phi\otimes f = \ssigma f$.
Moreover, again by Theorem~4.5 of~\cite{\BONN},
$f$ is not a form with complex multiplication.

The abelian variety $A=A_f$ associated to~$f$ satisfies
$\Q\otimes\End_\Q(A)=E$.  According to~\cite{\MATHANN} (see
especially Theorem~5.1 of~\cite{\MATHANN}), the full
algebra $\Cal X=\Q\otimes\End_{\Qbar}(A)$ of $A$ coincides with
its subalgebra $\Q\otimes\End_K(A)=E$, where $K=\Q(\sqrt{13})$
is the extension of~$\Q$ cut out by~$\phi$.  This algebra
is a quaternion algebra over~$\Q$.
Since $K$ may be embedded in~$\R$, a well known theorem of
Shimura~\cite{\BOUNDED, Th.~0} implies that $\Cal X$ is isomorphic
to the matrix algebra $M(2,\Q)$.
(Once we know that $E=\Q(\sqrt3)$, we can 
give an alternative proof that $\Cal X$
is a matrix algebra which is based on the
explicit description of~$\Cal X$
given in~\cite\MATHANN.)
Hence $A$ becomes isogenous over~$K$ to a product $C\times C$,
where $C$ is an elliptic curve defined over~$K$.  Therefore, we
are in the situation described above, with $E=\Q(\sqrt3)$
and $K=\Q(\sqrt{13})$.


\head 8. Descent of abelian varieties up to isogeny.%
\endhead\noindent
Suppose that $L/K$ is a Galois extension of~fields and
that $A$ is an abelian variety over~$L$.  A well-known
theorem of Weil \cite\WEILFD\ states
that $A$ has a model over $K$ if and only if there are
isomorphisms $\mu_\sigma\:\ssigma
A\buildrel\sim\over\to A$ ($\sigma\in\Gal(L/K)$)
which satisfy the compatibility condition
$$\mu_\sigma\ssigma\mu_\tau = \mu_{\sigma\tau}.\leqno(8.1)$$

As an application of the techniques encountered in~\S6, we will
prove an analogous criterion
for abelian varieties {\it up to isogeny}.
Namely, suppose that $A$ is isogenous (over~$L$) to an abelian
variety $B/L$ which has a model over~$K$.  Then one finds
isomorphisms $\mu_\sigma\:\ssigma A\buildrel\sim\over\to A$
of abelian varieties up to isogeny over~$L$ which satisfy the
compatibility condition~(8.1).  Conversely, one has
\proclaim{(8.2) Theorem}
Suppose that there are isomorphisms of abelian varieties up
to isogeny over~$L$,  $\mu\:\ssigma A\buildrel\sim\over\to A$,
which satisfy~(8.1).  Then there is an abelian variety $B$ over~$K$
such that $A$ is $L$-isogenous to $B_L$.
\endproclaim\startproof
%
We can, and do,
assume that $L$ is a finite extension of~$K$.
Let $X$ be the abelian variety~$X=\Res_{L/K} A$.
Recalling the discussion of~\S6, we find a decomposition
$$ \Q\otimes\End_K(X) = \prod_{\sigma\in\Gal(L/K)}
\Q\otimes\Hom_L(\ssigma A,A).$$
The homomorphism $\mu_\sigma$ in
the ``$\sigma^{\roman{th}}$ factor" corresponds to an
element $[\sigma]$ of~$\Q\otimes\End_K(X)$.
The element~$[g]$
operates on $\prod \ssigma A$ as
a matrix, sending ${}^{\tau g}\mkern-\thinmuskip A$ to
$\ttau A$ by $\ttau\mu_g$ for each $\tau\in\Gal(L/K)$.

Because
the analogue of~$c(\sigma,\tau)$ is~1 in this context, we have
simply $[\sigma][\tau]=[\sigma\tau]$ for $\sigma,\tau\in\Gal(L/K)$:
the identity~(8.1) shows that the product of the
matrices representing $[\sigma]$ and~$[\tau]$ is the matrix
representing $[\sigma\tau]$.
Hence $\Cal R\seteq\Q[\Gal(L/K)]$ operates on~$X$.

An analogue of Proposition~6.5 shows that we have
$$ X_L \approx \Cal R\otimes_\Q A $$
in the category of abelian varieties over~$L$ up to isogeny.
To see this explicitly, we let $A_\sigma$ be a copy of~$A$
indexed by $\sigma$ and consider the isomorphism of abelian
varieties over~$L$ up to isogeny
$$\iota\: \prod_\sigma A_\sigma \,\buildrel\sim\over\to\,
      \prod_\sigma \ssigma A = X_L$$
which takes $A_\sigma$ to ${}^{\sigma^{-1}}\mkern-\thinmuskip A$
via the map ${}^{\sigma^{-1}}\mkern-\thinmuskip \mu_\sigma$.
Via this isomorphism, the automorphism $[\sigma]$ of~$X$ is
transported to the permutation which takes each factor $A_g$ of
$\prod_g A_g$ to the factor $A_{\sigma g}$, via the identity map
$A\to A$.

Let $B$ be the image of $\eta\seteq \sum_\sigma [\sigma]$, so that
$B$ is an abelian subvariety of~$X$ which is defined
over~$K$.  (The sum $\eta$ need not
be a literal endomorphism of~$X$, so that, strictly speaking, one
should consider the image of a suitable multiple of~$\eta$.)
The isomorphism $\iota$ makes $B_L$ correspond
to the diagonal image of~$A$
in $\prod A_\sigma = A\times\cdots\times A$.  Hence $B_L$ is isogenous
to~$A$.
\endproof

\Refs

\catcode`\?=\active
\def?{.\hskip 0.1667em\relax}

\ref\no\ANTIV\by B?J. Birch and W. Kuyk, eds.
\book Modular functions of one variable IV
\bookinfo Lecture Notes in Math., vol. 476
\publ
   Spring-Verlag\publaddr Berlin and New York
\yr1975
\endref

\ref\no\GRBU\by J. Buhler and B?H. Gross
 \paper Arithmetic on elliptic curves with complex
 multiplication.~II\jour Invent. Math\yr1985\vol79\pages11--29\endref

\ref\no\CHI\by W. Chi
\paper Twists of central simple algebras and
endomorphism algebras of some abelian varieties
\jour Math. Ann.\vol276\yr1987\pages615--632\endref

\ref\no\DELCORVALIS\by P. Deligne
\paper Valeurs de fonctions $L$ et p\'eriodes d'int\'egrales
\jour Proceedings of Symposia in Pure Mathematics \vol 33\yr1979
\pages (2) 313--346\endref

\ref\no\FALT\by G. Faltings
\paper  Endlichkeitss\"atze f\"ur abelsche
    Variet\"aten \"uber Zahl\-k\"orpern
\jour Invent. Math.\vol73\yr1983\pages349--366\endref

\ref\no\FW\by G. Faltings, G. W\"ustholz et. al.
    \book Rational points
\publ
   F.~Vieweg \& Sohn
\publaddr Braunschweig/Wies\-baden \yr1984\endref

\ref\no\BHG\by B?H. Gross\book
Arithmetic on elliptic curves with complex multiplication
\bookinfo Lecture Notes in Math., vol. 776
\publ
   Spring-Verlag\publaddr Berlin and New York
\yr1980
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\ref\no\SGA\by A. Grothendieck  \paper SGA7 I, Expos\'e IX
\inbook Lecture Notes in Math., vol. 288
\publ   Springer-Verlag \publaddr Berlin and New York
\yr     1972 \pages 313--523
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\ref\no\HIDA\by H. Hida \paper On abelian varieties with
complex multiplication as factors of the Jacobians of
Shimura curves\jour Am. J. Math.\vol103\yr1981\pages727--776\endref

\ref\no\KOIKE\by M. Koike\paper On certain abelian varieties
  obtained from new forms of weight 2 on $\Gamma_0(3^4)$ and
  $\Gamma_0(3^5)$\jour Nagoya Math. J.\vol62\yr1976\pages29--39
  \endref

\ref\no\LORENZ\by D?J. Lorenzini\paper On the jacobian of the
modular curve $X_0(N)$\toappear\endref

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   B. Mazur \paper Number theory as gadfly
   \jour Am. Math. Monthly \vol98\yr1991\pages 593--610
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\ref\no\MILNE\by J?S. Milne\paper On the arithmetic
of abelian varieties\jour Invent. Math.\vol17\yr1972\pages
177--190\endref

\ref\no\MUMF\by D. Mumford\book Abelian varieties\bookinfo
Second edition\publ Oxford University Press\publaddr London\yr 1974
\endref

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ramification\jour Am. J. Math.\vol89\yr1967\pages1--21\endref


\ref\no\ANNALS\by K?A. Ribet \paper Endomorphisms of
semi-stable abelian varieties over number fields
\jour Annals of Math.\vol 101\yr1975\pages555--562\endref

\ref\no\MYTHESIS\bysame
  \paper
Galois action on division points on abelian
varieties with   many real    multiplications    
\jour Am. J.  Math. \vol 98 \yr1976\pages 751--804 
\endref

\ref\no\BONN\bysame
   \paper The $\ell$-adic representations attached to
  an eigenform  with Nebentypus: a survey
\inbook   Lecture Notes in Math., vol.  601
\publ   Springer-Verlag \publaddr Berlin and New York
\yr1977\pages 17--52\endref


\ref\no\MATHANN\bysame
\paper Twists of modular forms and endomorphisms of
abelian varieties\jour Math. Ann.\vol 253\yr1980\pages 43--62\endref

\ref\no\DPP\bysame
\paper Endomorphism algebras of Abelian varieties 
attached to newforms 
of weight~2\jour Prog\-ress in Math.\vol12\yr1981\pages
263--276\endref




\ref\no\MG\by J-P. Serre
    \book Abelian $\ell$-adic representations and elliptic curves
\publ
   Addison-Wesley Publ. Co.
\publaddr Redding, Mass. \yr1989\endref

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  des points d'ordre fini des courbes elliptiques\jour
  Invent. Math.\vol 15\yr1972\pages 259--331\endref


\ref\no\JPSDURHAM\bysame \paper Modular forms of weight one
  and Galois representations
  \paperinfo prepared in collaboration with C.~J.~Bushnell
\inbook Algebraic number fields ($L$-functions and Galois
properties) \bookinfo A.~Fr\"ohlich, ed.
\publ   Academic Press \publaddr London, New York and San Francisco
\yr     1977 \pages193--268
\endref

\ref\no\DUKE\bysame
 \paper  Sur les repr\'esentations modulaires de degr\'e  
   2 de $\GalQ$\jour  Duke Math. J.\vol 54\yr1987\pages 179--230\endref


\ref\no\DISCONT\by G. Shimura\paper Algebraic number fields and
  symplectic discontinuous groups \jour Ann. of Math.\vol86\yr1967
  \pages503--592\endref

\ref\no\PINK\bysame\paper On elliptic curves with complex
 multiplication as factors of the Jacobians of modular
 function fields\jour Nagoya Math. J.\vol43\yr1971\pages
 199--208\endref

\ref\no\SHIM\bysame
    \book Introduction to the arithmetic theory of automorphic functions
\publ
   Princeton University Press
\publaddr Princeton \yr1971\endref

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\paper Class fields over real quadratic fields and
Hecke operators\jour Ann. of Math.\vol95\yr1972\pages131--190\endref

\ref\no\FACTOR\bysame
  \paper On the factors of the Jacobian variety of a modular 
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\paper On the real points of an arithmetic quotient of
a bounded symmetric domain\jour Math. Ann.\vol215\yr1975
\pages 135--164\endref
\ref\no\SHTAN\by G. Shimura and Y. Taniyama
\book Complex multiplication of abelian varieties and its
 applications to number theory\bookinfo Second printing corrected
 \publ Math. Society of Japan \publaddr Tokyo \yr1975\endref

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The field of definition of a variety\jour Am. J. Math\vol 78\yr1956
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\ref\no\WEIL\bysame\book Adeles and algebraic groups
\bookinfo  Progress in Math., vol. 23
\publ Birkh\"auser
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\endRefs
\enddocument
